On pairs of triangular numbers whose product is a perfect square and pairs of intervals of successive integers with equal sums of squares

TL;DR

This paper explores properties of triangular numbers, quadratic numbers, and their relationships to perfect squares, constructing polynomial solutions and conjecturing their completeness in covering all quadratic squares, while also examining equal sums of squares in integer intervals.

Contribution

It introduces polynomial constructions for pairs of integers related to square quadratic numbers and conjectures their comprehensive coverage of all such squares, also analyzing intervals with equal sums of squares.

Findings

Constructed polynomial solutions for square quadratic numbers.

Conjectured that these solutions cover all quadratic squares.

Identified pairs of successive integer intervals with equal sums of squares.

Abstract

A number $N$ is a triangular number if it can be written as $N = t(t + 1)/2$ for some nonnegative integer number $t$. A triangular number $N$ is called square if it is a perfect square, that is, $N = d^2$ for some integer number $d$. Square triangular numbers were characterized by Euler in 1778 and are in one-to-one correspondence with the so-called near-isosceles Pythagorean triples $(k,k+1,l)$, where $k^2 + (k+1)^2 = l^2$. A quadratic number is the product $\Pi = \Pi(k,j) = k(k+1)(k+j)(k+j+1)$ for some nonnegative integer numbers $k$ and $j$. By definition, it is the product of two triangular numbers and 4. Quadratic number $\Pi$ and the corresponding pair $(k,j)$ are called square if $\Pi$ is a perfect square. Clearly, $(k,j)$ is square if both triangular numbers $k(k+1)/2$ and $(k+j)(k+j+1)/2$ are perfect squares. Yet, there exist infinitely many other square quadratic numbers. We construct polynomials $j_i(k)$ of degree $i$ with positive integer coefficients satisfying equations: $k + j_{2 \ell}(k) + 1 = k [a_\ell k^\ell + \dots + a_1 k + a_0]^2 +1 = (k+1) [b_\ell k^\ell + \dots + b_1 k + b_0]^2$ and \newline $k + j_{2\ell+1}(k) + 1 = k(k+1) [a_\ell k^\ell + \dots + a_1 k + a_0]^2 + 1 = [b_{\ell+1} k^{\ell+1}+b_\ell k^\ell + \dots + b_1 k + b_0]^2$ for some positive integer $\ell$ and some coefficients $a_i, b_j$, $i=0, \ldots, \ell, j=0, \ldots, \ell+1$. All the obtained pairs $(k, j_i(k))$ are square. We conjecture that the products of square triangular numbers and pairs $(k, j_i(k))$ cover all quadratic squares. Additionally, we identify pairs of intervals of successive integers with equal sums of squares.